In his book page 250 Exer 11:
Suppose that $I=[a,b]$,$\Omega$ is a region ,$I\subset\Omega$,$f$ is continuous in $\Omega$,and $f\in H(\Omega-I)$,prove that actually $f\in H(\Omega)$.
If I follow the ...

If I have an entire function $\phi$ such that it is of exponential order zero. I.e for all $\rho > 0$ we get $|\phi(s)|\le C_\rho e^{|s|^{\rho}}$. Furthermore, I have an extreme decay in the Taylor ...

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$.
Suppose we have the following representation:
...

Let $Z$ denote a subset of $\mathbb{C}$. Then some functions $f : Z \rightarrow \mathbb{C}$ have the property that there exist sequences $a,b : \mathbb{N} \rightarrow \mathbb{C}$ such that for all $z ...

So I was reading about Laurent series and $\mathrm{e}^{-1/x^2}$ was used as an example. We define the function $f(x) = \mathrm{e}^{-1/x^2}$ for $x \neq 0$ and $f(0)=0$. Then it was stated that, as a ...

Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...

Let $r\in\mathbb N$ and $f$ be an entire function on $\mathbb C$ such that for every $R\in\mathbb C[z]$, there exist polynomials $P_{i,R}(z)\in\mathbb{C}[z]$ ($0\le i\le r$) not all zero such that, ...

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...

Being inspired by this post, I've wondered if the infinite series below may be expressed as
an intregral. I'm very curious about that.
$$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$
...

Consider the function
$$f(z)=\frac{e^{\frac{1}{z-1}}}{e^z -1}$$
$z_0=1$ is an essential singularity, hence
$$f(z)=\displaystyle\sum_{-\infty}^{+\infty}a_n(z-1)^n$$
near to $z_0=1$ and I want to find ...

Let $X$ be a complex manifold with a Hermitian metric. Is there a "complex" analogue of geodesics on $X$ which is of any interest? For example, is anything known about holomorphic maps $f : \mathbb C ...

I would like to prove that if a function $f(z)$ is holomorphic on $\overline{D(P,r)}$ and one to one on $\partial D(P,r)$ then $f$ is one to one on $D(P,r)$. I noticed that for $w \in f(D(P,r))$ with ...

I was trying to prove that for a standard complex Gaussian variable $Z$ it holds that $|Z|^2$ is exponentially distributed with parameter 1, $\frac{Z}{|Z|}$ is uniformly distributed on the unit circle ...